Strong Gohberg–Markus–Boltyanski–Hadwiger conjecture

Let $d\geq 1$, and let $K$ be a convex body in $\mathbb R^d$, meaning a compact convex set with non-empty interior. A translate of $\lambda K$, with $0<\lambda<1$, is a smaller positive homothetic copy of $K$. Strong Gohberg–Markus–Boltyanski–Hadwiger conjecture. For every $d\geq 1$, there is a number $0<\lambda_d<1$ such that every convex body $K$ in $\mathbb R^d$ is covered by $2^d$ translates of $\lambda_d K$.

This is called formally stronger because the homothety ratio depends only on the dimension, not on the convex body. The preceding theorem in the paper shows that this uniform statement is equivalent to the ordinary Gohberg–Markus–Boltyanski–Hadwiger conjecture, and the supplied text gives no resolution.